Reduced model of macro-scale stochastic plasticity identification by Bayesian inference: Application to quasi-brittle failure of concrete

Abstract

In this paper we deal with a probability-based scale bridging for concrete material when passing the detailed information at the meso-scale (the scale where the aggregate vs. cement microstructure is visible) with a Voronoi-cell based microstructure representation towards the chosen reduced model at the macro-scale (the scale where the concrete is a homogenized continuum) with a stochastic plasticity model for localized failure. This is accomplished by using Bayesian inference providing the probability distributions of the macro-scale model parameters expressed as random variables (RV) in order to compensate for the model reduction from the meso-scale, where parameters are expressed as random fields (RF). The original aspect of this approach is in the resulting macro-scale stochastic plasticity model, which can best quantify the uncertainty due to data loss in terms of the corresponding probability distribution of its parameters. The proposed procedure is illustrated in detail for the concrete meso-scale model presented in Part I of this paper (see Karavelić et al. (2019)), both for the simple elastic response, as well as for the plastic response with hardening in the fracture process zone (FPZ), followed by a softening response in the localized failure phase. This context of localized failure implies that the classical homogenization procedure no longer applies, and should be replaced by a macro-scale reduced model defined with respect to a quantity of interest (QoI), which is not necessarily the same for each particular response phase. The complete set of results for the parameter identification is combined together at the macro-scale in terms of a solid finite element with embedded discontinuity (ED-FEM), granting it very powerful predictive properties. The particular choice of ED-FEM for the macro-scale reduced model allows one to compute the QoI at an element-level, either as strain energy for the elastic response, or as plastic dissipation for each failure mode. Thus, the Bayesian inference computation of the probability distributions of the RVs representing the macro-scale model parameters, performed by matching such QoI results computed for the macro-scale and the meso-scale models, reduces to a homogenization-type procedure within a probabilistic setting that can successfully handle any case where the separation of scales no longer applies.